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Ratio and Proportion (Including Sharing in a Ratio) · ICSE STD 7 (ICSE - English Medium). 24 questions.

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Question 13 Marks
Show that 2, 12 and 72 are in continued proportion.
Answer
Three numbers $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are in continued
proportion if, $a: b:: b: c$
The numbers are 2, 12 and 72
$ \begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{2}{12}=\frac{1}{6} \\ & \frac{\mathrm{b}}{\mathrm{c}}=\frac{12}{72}=\frac{1}{6} \end{aligned} $
As, $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{b}}{\mathrm{c}}$
$\therefore 2,12$ and 72 are in continued proportion.
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Question 23 Marks
Find the value of x, when 2.5 : 4 = x : 7.5.
Answer

$\begin{aligned} & 2.5: 4=x: 7.5 \\ & 4 \times x=2.5 \times 7.5 \\ & x=\frac{2.5 \times 7.5}{4} \\ & x=\frac{25 \times 75}{4 \times 100} \\ & x=\frac{75}{16}=4 \frac{11}{16}\end{aligned}$
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Question 33 Marks
If x: y - 5 :4 and 2 : x = 3 :8, find the value of y.
Answer
$x: y=5: 4$
and $2: x=3: 8$
Then, $\frac{\mathrm{x}}{\mathrm{y}}=\frac{5}{4}$...(i)
and $\frac{2}{\mathrm{x}}=\frac{3}{8}$...(ii)
$\mathrm{x}=\frac{2 \times 8}{3}=\frac{16}{3}$
Now put the value of $x$ in eq. (i)
$ \frac{x}{y}=\frac{5}{4} $
$y=x \times \frac{4}{5}$
$ y=\frac{16}{3} \times \frac{4}{5}=\frac{64}{15} $
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Question 43 Marks
If a : b = 1.5 : 3.5 and b : c = 5 : 6, find a : c.
Answer

$\begin{aligned} & \mathrm{a}: \mathrm{b}=1.5: 3.5 \\ & \frac{\mathrm{a}}{\mathrm{b}}=\frac{1.5}{3.5} \frac{15}{35}=\frac{3}{7} \\ & \mathrm{~b}: \mathrm{c}=5: 6 \\ & \therefore \frac{\mathrm{b}}{\mathrm{c}}=\frac{5}{6} \\ & \text { Now } \frac{a}{b} \times \frac{b}{c}=\frac{3}{7} \times \frac{5}{6}=\frac{5}{14} \\ & \therefore \frac{a}{c}=\frac{5}{14} \\ & \Rightarrow \mathrm{a}: \mathrm{c}=5: 14\end{aligned}$
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Question 53 Marks
If $P: Q=\frac{1}{2}: \frac{1}{3}$ and $Q: R=1 \frac{1}{2}: 1 \frac{1}{3}$, find $P: R$.
Answer

$\begin{aligned} & P: Q=\frac{1}{2}: \frac{1}{3} \\ & \therefore \frac{P}{Q}=\frac{1}{2} \times \frac{3}{1}=\frac{3}{2} \\ & \text { and } Q: R=1 \frac{1}{2}: 1 \frac{1}{3}=\frac{3}{2}: \frac{4}{3} \\ & \therefore \frac{Q}{R}=\frac{3}{2} \times \frac{3}{4}=\frac{9}{8} \\ & \text { Now } \frac{P}{Q} \times \frac{Q}{R}=\frac{3}{2} \times \frac{9}{8} \\ & \Rightarrow \frac{P}{R}=\frac{27}{16} \\ & \therefore P: R=27: 16\end{aligned}$
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Question 63 Marks
If m : n = 4 : 9 and n : s : 3 : 7, find m : s
Answer

$\begin{aligned} & \mathrm{m}: \mathrm{n}=4: 9 \\ & \frac{\mathrm{m}}{\mathrm{n}}=\frac{4}{9} \\ & \text { and } \mathrm{n}: \mathrm{s}: 3: 7 \\ & \therefore \frac{\mathrm{n}}{\mathrm{s}}=\frac{3}{7} \\ & \therefore \frac{\mathrm{m}}{\mathrm{n}} \times \frac{\mathrm{n}}{\mathrm{s}}=\frac{4}{9} \times \frac{3}{7} \\ & \mathrm{~m}: \mathrm{s}=4: 21\end{aligned}$
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Question 73 Marks
If x : y = 2 : 3 and y : z = 5 : 7, find x : y : z.
Answer

$\begin{aligned} & x: y=2: 3 \\ & \left.=\frac{2}{3}: 1 \quad \text { (Dividing by } 3\right.) \\ & \text { and } y: z=5: 7 \\ & =1: \frac{7}{5} \quad \ldots(\text { Dividing by } 5) \\ & \therefore x: y: z=\frac{2}{3}: 1: \frac{7}{5} \\ & =10: 15: 21 \quad \ldots(\text { Multiplying by } 3 \times 5=15 \text { ) }\end{aligned}$
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Question 83 Marks
If A : B = 3 : 5 and B : C = 4 : 7, find A : B : C
Answer

$\begin{aligned} & A: B=3: 5 \\ & \left.=\frac{3}{5}: 1 \quad \text { (Dividing by } 5\right.)\\ & \text { and } B: C=4: 7 \\ & =1: \frac{7}{4} \quad \ldots(\text { Dividing by } 4) \\ & \therefore A: B: C=\frac{3}{5}: 1: \frac{7}{4} \\ & =12: 20: 35 \quad \ldots(\text { Multiplying by } 5 \times 4=20)\end{aligned}$
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Question 93 Marks
Find the mean proportional between 0.6 and 9.6
Answer
Mean proportional between 0.6 and 9.6
$ \begin{aligned} & =\sqrt{0.6 \times 9.6} \\ & =\sqrt{\frac{6}{10} \times \frac{96}{10}} \\ & =\sqrt{\frac{576}{100}}=\frac{24}{10} \\ & =2.4 \end{aligned} $
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Question 103 Marks
Check whether the following quantities form a proportion or not? $2 \frac{1}{2}, 5 \frac{1}{2}, 3.0$ and 6.0
Answer
$2 \frac{1}{2}, 5 \frac{1}{2}, 3.0$ and 6.0 are in proportion
$ \begin{aligned} & \text { if } 2 \frac{1}{2} \times 6.0=5 \frac{1}{2} \times 3.0 \\ & \Rightarrow \frac{5}{2} \times 6.0=\frac{11}{2} \times 3.0 \Rightarrow \frac{30}{2}=\frac{33}{2} \end{aligned} $
which is not true.
Hence $2 \frac{1}{2}, 5 \frac{1}{2}, 3.0$ and 6.0 are not in proportion.
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Question 113 Marks
Check whether the following quantities form a proportion or not?
0.4, 0.5, 2.9 and 3.5
Answer
0.4, 0.5, 2.9 and 3.5 are in proportion
if 0.4 × 3.5 = 0.5 × 2.9
⇒ 1.40 = 1.45
which is not true
Hence 0.4, 0.5, 2.9 and 3.5 are not in proportion.
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Question 123 Marks
Check whether the following quantities form a proportion or not?
$1 \frac{1}{2}, 3 \frac{1}{4}, 4 \frac{1}{2} \text { and } 9 \frac{3}{4}$
Answer
$1 \frac{1}{2}, 3 \frac{1}{4}, 4 \frac{1}{2}$ and $9 \frac{3}{4}$ are in proportion
if $1 \frac{1}{2} \times 9 \frac{3}{4}=3 \frac{1}{4} \times 4 \frac{1}{2}$
$\Rightarrow \frac{3}{2} \times \frac{39}{4}=\frac{13}{4} \times \frac{9}{2}$
$\Rightarrow \frac{117}{8}=\frac{117}{8}$ which is true
Hence $1 \frac{1}{2}, 3 \frac{1}{4}, 4 \frac{1}{2}$ and $9 \frac{3}{4}$ are in proportion.
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Question 133 Marks
Check whether the following quantities form a proportion or not?
0.8, 3, 2.4 and 9
Answer
0.8, 3, 2.4 and 9 are in proportion.
if 0.8 × 9 = 3 × 2.4
⇒ 7.2 = 7.2
which is true
Hence 0.8, 3, 2.4 and 9 are in proportion.
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Question 143 Marks
Check whether the following quantities form a proportion or not?
3x, 7x, 24 and 56
Answer
3x, 7x, 24 and 56
If these are in proportion, then
3x × 56 = 7x × 24
⇒ 168 x = 168 x
which is true.
Hence 3x, 7x, 24 and 56 are in proportion.
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Question 153 Marks
Two numbers are in the ratio 10: 11. Their sum is 168. Find the numbers.
Answer
Ratio between two numbers $=10: 11$
Sum of ratios $=10+11=21$
Total sum $=168$
$\therefore$ First number $=\frac{168}{21} \times 10=80$
Second number $=\frac{168}{21} \times 11=88$
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Question 163 Marks
Two numbers are in the ratio 5: 7. Their difference is 10. Find the numbers.
Answer
Ratio between two numbers $=5: 7$
Difference $=7-5=2$
If difference is 2 , then first number $=5$
and if difference is 10 , then first number
$ =\frac{5}{2} \times 10=25 $
and second number $=\frac{7}{2} \times 10=35$
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Question 173 Marks
A rectangular field is 100 m by 80 m. Find the ratio of breadth to its perimeter.
Answer
Length of field (l) = 100 m
Breadth (b) = 80 m
∴Perimeter = 2 (l + b) = 2 (100 + 80) m = 2 x 180 = 360 m
Ratio between breadth and its perimeter
= 80 : 360 = 2 : 9
(Dividing by 40, the HCF of 80 and 360)
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Question 183 Marks
A rectangular field is 100 m by 80 m. Find the ratio of length to its breadth.
Answer
Length of field (l) = 100 m
Breadth (b) = 80 m
∴Perimeter = 2 (l + b) = 2 (100 + 80) m = 2 x 180 = 360 m
Ratio between length and breadth
= 100 : 80 = 5 : 4
(Dividing by 20, the HCF of 100 and 80)
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Question 193 Marks
The angles of a triangle are in the ratio 3 :2 : 7. Find each angle.
Answer
Ratio in angles of a triangle $=3: 2: 7$
Sum of ratios $=3+2+7=12$
Sum of angles of a triangle $=180^{\circ}$
$\therefore$ First angle $=\frac{3}{12} \times 180^{\circ}=45^{\circ}$
Second angle $=\frac{2}{12} \times 180^{\circ}=30^{\circ}$
Third angle $=\frac{7}{12} \times 180^{\circ}=105^{\circ}$
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Question 203 Marks
The ratio between the prices of a scooter and a refrigerator is 4 : 1. If the scooter costs ₹ 45,000 more than the refrigerator, find the price of the refrigerator.
Answer
Ratio between the prices of scooter and a refrigerator $=4: 1$
Cost price of scooter ₹ 45,000
Let the cost of scooter $=4 x$
Cost of refrigerator $=1 \mathrm{x}$
According to condition,
Cost of scooter $>$ Cost of refrigerator
$ \begin{aligned} & \Rightarrow 4 \mathrm{x}-1 \mathrm{x}=45000 \\ & \Rightarrow 3 \mathrm{x}=45000 \end{aligned} $
$ x=\frac{45000}{3} $
$ \Rightarrow x=15000 $
$\therefore$ Price of refrigerator ₹ 15000
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Question 213 Marks
Is divided between A and B in such a way that A gets half of B. Find :
(i) the ratio between the shares of A and B.
(ii) the share of A and the share of B.
Answer
Total amount to be divided between A and B ₹ 300
(i) A gets half of $B$
Hence, ratio between $A$ and $B=\frac{1}{2}=1: 2$
(ii) Sum of ratios $=1+2=3$
$ \begin{aligned} & \therefore A^{\prime} \text { shares }=\frac{300 \times 1}{3}=100 \\ & \therefore B^{\prime} \text { share }=\frac{300 \times 2}{3}=200 \end{aligned} $
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Question 223 Marks
Ten gram of an alloy of metals A and B contains 7.5 gm of metal A and the rest is metal B. Find the ratio between :
(i) the weights of metals A and B in the alloy.
(ii) the weight of metal B and the weight of the alloy.
Answer
Total weight of A and B metals $=10 \mathrm{gm} A^{\prime} \mathrm{s}$ weight $=7.5 \mathrm{gm} B^{\prime}$ s weight $=10-7.5=2.5 \mathrm{gm}$
(i) Ratio between $A$ and $B=7.5: 2.5$
$ =\frac{75}{10}: \frac{25}{10}=3: 1 $
(ii) Ratio between B and total alloy
$ \begin{aligned} & =2.5: 10=\frac{25}{10}: 10 \\ & \Rightarrow 25: 100=1: 4 \end{aligned} $
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Question 233 Marks
The population of the town is 180,000 , out of which males are $\frac{1}{3}$ of the whole population. Find the number of females. Also, find the ratio of the number of females to the whole population.
Answer
Total population $=1,80,000$
Population of males $=\frac{1}{3} \times 1,80,000=60,000$
$\therefore$ Population of females $=1,80,000-60,000=1,20,000$
The ratio of females to the whole population
$ =1,20,000: 1,80,000=12: 18=2: 3 $
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Question 243 Marks
A line is divided into two parts in the ratio 2.5 : 1.3. If the smaller one is 35.1 cm, find the length of the line.
Answer
Ratio between two part $=2.5: 1.3$
Let first part $=2.5 \mathrm{x} \mathrm{cm}$ and second part $=1.3 x \mathrm{~cm}$
$ 1.3 x=35.1 $
$ x=\frac{35.1}{1.3} $
$ x=27 $
Length of larger part $=2.5 \times 27$
$ =67.5 \mathrm{~cm} $
Length of line $=35.1+67.5$
$ =102.6 \mathrm{~cm} $
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[3 marks sum] - MATHS STD 7 Questions - Vidyadip